J-holomorphic curve

For almost complex manifolds $(\Sigma ,j)$ and $(X,J)$, a smooth map $u:\Sigma \to X$ is J-holomorphic when $du\circ j=J\circ du$, equivalently ${\bar{\partial }}_{J}u=0$.

For compatible metrics on a Riemann surface and symplectic target,Thus a J-holomorphic curve minimizes energy among maps in its homology class.

On a compact region with controlled compatible almost complex structure, there are $c,r_0>0$ such that a nonconstant J-holomorphic curve through the center of a ball of radius $r<r_0$, with boundary outside the ball, has symplectic area at least $c{r}^2$. For the standard complex structure the sharp local constant is $\pi$.